| L(s) = 1 | − 32·4-s + 52·7-s − 20·13-s + 640·16-s + 100·19-s + 1.70e3·25-s − 1.66e3·28-s + 2.95e3·31-s − 32·37-s + 136·43-s − 1.06e4·49-s + 640·52-s + 8.95e3·61-s − 1.02e4·64-s − 1.50e4·67-s + 2.07e4·73-s − 3.20e3·76-s + 1.21e4·79-s − 1.04e3·91-s − 6.26e4·97-s − 5.45e4·100-s + 2.66e3·103-s + 3.11e4·109-s + 3.32e4·112-s + 3.07e4·121-s − 9.45e4·124-s + 127-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.06·7-s − 0.118·13-s + 5/2·16-s + 0.277·19-s + 2.72·25-s − 2.12·28-s + 3.07·31-s − 0.0233·37-s + 0.0735·43-s − 4.45·49-s + 0.236·52-s + 2.40·61-s − 5/2·64-s − 3.34·67-s + 3.88·73-s − 0.554·76-s + 1.93·79-s − 0.125·91-s − 6.66·97-s − 5.45·100-s + 0.251·103-s + 2.62·109-s + 2.65·112-s + 2.10·121-s − 6.15·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{32}\right)^{s/2} \, \Gamma_{\C}(s+2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.204167549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.204167549\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 1706 T^{2} + 1350097 T^{4} - 1129364066 T^{6} + 885462580036 T^{8} - 1129364066 p^{8} T^{10} + 1350097 p^{16} T^{12} - 1706 p^{24} T^{14} + p^{32} T^{16} \) |
| 7 | \( ( 1 - 26 T + 6361 T^{2} - 72938 T^{3} + 18730420 T^{4} - 72938 p^{4} T^{5} + 6361 p^{8} T^{6} - 26 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 11 | \( 1 - 30764 T^{2} + 719532634 T^{4} - 11874523481552 T^{6} + 193764526125181411 T^{8} - 11874523481552 p^{8} T^{10} + 719532634 p^{16} T^{12} - 30764 p^{24} T^{14} + p^{32} T^{16} \) |
| 13 | \( ( 1 + 10 T + 205 p^{2} T^{2} - 3833642 T^{3} + 1085929144 T^{4} - 3833642 p^{4} T^{5} + 205 p^{10} T^{6} + 10 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 17 | \( 1 - 249218 T^{2} + 30482460289 T^{4} - 3595961342423810 T^{6} + \)\(36\!\cdots\!56\)\( T^{8} - 3595961342423810 p^{8} T^{10} + 30482460289 p^{16} T^{12} - 249218 p^{24} T^{14} + p^{32} T^{16} \) |
| 19 | \( ( 1 - 50 T + 226153 T^{2} - 12428354 T^{3} + 33132078964 T^{4} - 12428354 p^{4} T^{5} + 226153 p^{8} T^{6} - 50 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 23 | \( 1 - 1677650 T^{2} + 1331124941041 T^{4} - 655538322775349810 T^{6} + \)\(21\!\cdots\!20\)\( T^{8} - 655538322775349810 p^{8} T^{10} + 1331124941041 p^{16} T^{12} - 1677650 p^{24} T^{14} + p^{32} T^{16} \) |
| 29 | \( 1 - 2794826 T^{2} + 3890007513841 T^{4} - 3549662430566591234 T^{6} + \)\(26\!\cdots\!00\)\( T^{8} - 3549662430566591234 p^{8} T^{10} + 3890007513841 p^{16} T^{12} - 2794826 p^{24} T^{14} + p^{32} T^{16} \) |
| 31 | \( ( 1 - 1478 T + 2458093 T^{2} - 1823889458 T^{3} + 2106691989592 T^{4} - 1823889458 p^{4} T^{5} + 2458093 p^{8} T^{6} - 1478 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 37 | \( ( 1 + 16 T + 5152060 T^{2} + 1816976752 T^{3} + 11981316770374 T^{4} + 1816976752 p^{4} T^{5} + 5152060 p^{8} T^{6} + 16 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 41 | \( 1 - 21961436 T^{2} + 212758908574570 T^{4} - \)\(11\!\cdots\!84\)\( T^{6} + \)\(41\!\cdots\!19\)\( T^{8} - \)\(11\!\cdots\!84\)\( p^{8} T^{10} + 212758908574570 p^{16} T^{12} - 21961436 p^{24} T^{14} + p^{32} T^{16} \) |
| 43 | \( ( 1 - 68 T + 10753162 T^{2} + 662295184 T^{3} + 50299570203427 T^{4} + 662295184 p^{4} T^{5} + 10753162 p^{8} T^{6} - 68 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 47 | \( 1 - 27743762 T^{2} + 375094997142097 T^{4} - \)\(31\!\cdots\!50\)\( T^{6} + \)\(18\!\cdots\!72\)\( T^{8} - \)\(31\!\cdots\!50\)\( p^{8} T^{10} + 375094997142097 p^{16} T^{12} - 27743762 p^{24} T^{14} + p^{32} T^{16} \) |
| 53 | \( 1 - 47149064 T^{2} + 1030375014657436 T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!10\)\( T^{8} - \)\(13\!\cdots\!20\)\( p^{8} T^{10} + 1030375014657436 p^{16} T^{12} - 47149064 p^{24} T^{14} + p^{32} T^{16} \) |
| 59 | \( 1 - 47906492 T^{2} + 1355076391006186 T^{4} - \)\(25\!\cdots\!40\)\( T^{6} + \)\(36\!\cdots\!15\)\( T^{8} - \)\(25\!\cdots\!40\)\( p^{8} T^{10} + 1355076391006186 p^{16} T^{12} - 47906492 p^{24} T^{14} + p^{32} T^{16} \) |
| 61 | \( ( 1 - 4478 T + 20178421 T^{2} - 107824319498 T^{3} + 492507561368584 T^{4} - 107824319498 p^{4} T^{5} + 20178421 p^{8} T^{6} - 4478 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 67 | \( ( 1 + 112 p T + 78657610 T^{2} + 408157367920 T^{3} + 2385910495071643 T^{4} + 408157367920 p^{4} T^{5} + 78657610 p^{8} T^{6} + 112 p^{13} T^{7} + p^{16} T^{8} )^{2} \) |
| 71 | \( 1 - 125290160 T^{2} + 8106610376011420 T^{4} - \)\(34\!\cdots\!88\)\( T^{6} + \)\(10\!\cdots\!18\)\( T^{8} - \)\(34\!\cdots\!88\)\( p^{8} T^{10} + 8106610376011420 p^{16} T^{12} - 125290160 p^{24} T^{14} + p^{32} T^{16} \) |
| 73 | \( ( 1 - 10358 T + 106803217 T^{2} - 769838088062 T^{3} + 4439859073965124 T^{4} - 769838088062 p^{4} T^{5} + 106803217 p^{8} T^{6} - 10358 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 79 | \( ( 1 - 6050 T + 78749305 T^{2} - 104223044090 T^{3} + 2248951397883988 T^{4} - 104223044090 p^{4} T^{5} + 78749305 p^{8} T^{6} - 6050 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
| 83 | \( 1 - 313035410 T^{2} + 45550125871994641 T^{4} - \)\(40\!\cdots\!50\)\( T^{6} + \)\(23\!\cdots\!00\)\( T^{8} - \)\(40\!\cdots\!50\)\( p^{8} T^{10} + 45550125871994641 p^{16} T^{12} - 313035410 p^{24} T^{14} + p^{32} T^{16} \) |
| 89 | \( 1 - 206196488 T^{2} + 19314405141094684 T^{4} - \)\(14\!\cdots\!60\)\( T^{6} + \)\(10\!\cdots\!86\)\( T^{8} - \)\(14\!\cdots\!60\)\( p^{8} T^{10} + 19314405141094684 p^{16} T^{12} - 206196488 p^{24} T^{14} + p^{32} T^{16} \) |
| 97 | \( ( 1 + 31336 T + 674173114 T^{2} + 9488334796720 T^{3} + 104492670281518747 T^{4} + 9488334796720 p^{4} T^{5} + 674173114 p^{8} T^{6} + 31336 p^{12} T^{7} + p^{16} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.98103382609694280067169591572, −4.89540298215154143902779316783, −4.82906618226129796754133799985, −4.68129096615400599784408613827, −4.49623769311208230456874151932, −4.46366639481279315021168526199, −4.21584671480503764340590577927, −3.86584166253146773343423183652, −3.75959997750106543312901029340, −3.74810542436469304987286512161, −3.24358084593751321010520433830, −3.09636820111863632900552269742, −3.05911015388944532553286860334, −2.99626925872991600251434512979, −2.65611830107352246187837431932, −2.37995325472130210994942675320, −2.11587358883942276072010627517, −1.73530417957589550058323192027, −1.71790736709338130995094168727, −1.33135900645001101206538458991, −1.14379141004344027537358435250, −0.801420982285584519057414226800, −0.70956239824640340872468515086, −0.59137849122832808753372789736, −0.10080739070977426889023613241,
0.10080739070977426889023613241, 0.59137849122832808753372789736, 0.70956239824640340872468515086, 0.801420982285584519057414226800, 1.14379141004344027537358435250, 1.33135900645001101206538458991, 1.71790736709338130995094168727, 1.73530417957589550058323192027, 2.11587358883942276072010627517, 2.37995325472130210994942675320, 2.65611830107352246187837431932, 2.99626925872991600251434512979, 3.05911015388944532553286860334, 3.09636820111863632900552269742, 3.24358084593751321010520433830, 3.74810542436469304987286512161, 3.75959997750106543312901029340, 3.86584166253146773343423183652, 4.21584671480503764340590577927, 4.46366639481279315021168526199, 4.49623769311208230456874151932, 4.68129096615400599784408613827, 4.82906618226129796754133799985, 4.89540298215154143902779316783, 4.98103382609694280067169591572
Plot not available for L-functions of degree greater than 10.
